3.1 Basic Amplifier Configurations

The most fundamental application of the operational amplifier is, as its name suggests, amplification. By employing a feedback network—typically composed of resistors—the Op-Amp’s extremely high open-loop gain can be precisely controlled to create stable, predictable, and highly useful amplifier circuits. The three primary configurations are the inverting amplifier, the non-inverting amplifier, and the voltage follower.

A key concept used in analyzing these circuits is the virtual short. This phenomenon is a direct consequence of the nearly infinite open-loop gain (A_v) discussed in Section 2.2. When an Op-Amp is configured with negative feedback, the feedback loop forces the output voltage to a value that makes the differential input voltage (v_{inverting} – v_{non-inverting} = v_0 / A_v) approach zero. This means the voltage at the inverting terminal is effectively equal to the voltage at the non-inverting terminal, even though there is no physical connection between them.

Inverting Amplifier

An inverting amplifier produces an output that is an amplified version of the input, but with its polarity inverted (a 180° phase shift). The input signal is applied to the inverting terminal through a resistor R_1, and a feedback resistor R_f is connected between the output and the inverting terminal. The non-inverting terminal is connected to ground.

Gain Derivation:

1. The non-inverting terminal is at ground potential (0V).
2. Due to the virtual short concept, the voltage at the inverting terminal is also 0V.
3. We can write the nodal equation at the inverting terminal by summing the currents entering and leaving the node: \frac{0-V_i}{R_1}+ \frac{0-V_0}{R_f}=0
4. Rearranging the terms: \frac{-V_i}{R_1}= \frac{V_0}{R_f}
5. Solving for the output voltage, V_0: V_{0}=\left(\frac{-R_f}{R_1}\right)V_{i}

The voltage gain (A_v = V_0 / V_i) of the inverting amplifier is therefore:

\frac{V_0}{V_i}= \frac{-R_f}{R_1}

The negative sign in the gain equation is significant; it indicates the 180° phase difference between the input and the output. The magnitude of the gain is determined solely by the ratio of the feedback resistor to the input resistor.

Non-Inverting Amplifier

A non-inverting amplifier produces an amplified output that is in phase with the input signal. In this configuration, the input signal V_i is applied directly to the non-inverting terminal. The feedback network is identical to that of the inverting amplifier.

Gain Derivation:

1. The voltage at the non-inverting terminal is equal to the input voltage, V_i.
2. Due to the virtual short concept, the voltage at the inverting terminal is also equal to V_i.
3. Using the voltage division principle for the feedback network, the voltage at the inverting terminal (V_1) can be expressed in terms of the output voltage V_0: V_{1} = V_{0}\left(\frac{R_1}{R_1+R_f}\right)
4. Since V_1 = V_i, we can set the equations equal: V_{0}\left(\frac{R_1}{R_1+R_f}\right)=V_{i}
5. Solving for the voltage gain (A_v = V_0 / V_i): \frac{V_0}{V_i}=\frac{R_1+R_f}{R_1}

This simplifies to the final gain equation:

\frac{V_0}{V_i}=1+\frac{R_f}{R_1}

The positive sign indicates that there is no phase difference between the input and output. The gain is always greater than or equal to one.

Voltage Follower

A voltage follower is a special case of the non-inverting amplifier that provides a unity voltage gain. Its output voltage is exactly equal to its input voltage. It is created by setting the feedback resistor R_f to zero and the input resistor R_1 to infinity (an open circuit). In practice, this means the output is connected directly back to the inverting input.

Gain Derivation:

1. The input voltage V_i is applied to the non-inverting terminal.
2. The output voltage V_0 is connected directly to the inverting terminal.
3. According to the virtual short concept, the voltages at the two terminals are equal: V_{0} = V_{i}

The voltage gain is exactly one. The primary utility of a voltage follower is not to amplify voltage but to act as a buffer. Its very high input impedance prevents it from drawing current from the source, and its very low output impedance allows it to drive a load without voltage loss. It effectively isolates the source from the load.

These fundamental amplifier configurations form the basis for more complex circuits that can perform mathematical operations.

3.2 Arithmetic Circuits

Beyond simple amplification, operational amplifiers can be configured to perform fundamental arithmetic operations such as addition and subtraction. These circuits are the foundational elements of analog computing and are widely used in signal processing applications.

Adder (Summing Amplifier)

An adder, or summing amplifier, is a circuit that produces an output voltage equal to the weighted sum of its input voltages. It is a modification of the inverting amplifier configuration, with multiple input resistors connected to the inverting terminal.

For a two-input adder, voltages V_1 and V_2 are applied through resistors R_1 and R_2, respectively, to the inverting terminal. The non-inverting terminal is grounded.

Output Voltage Derivation:

1. With the non-inverting terminal at 0V, the virtual short concept places the inverting terminal at 0V as well.
2. The nodal equation at the inverting terminal is the sum of the currents from each input and the feedback path: \frac{0-V_1}{R_1}+\frac{0-V_2}{R_2}+\frac{0-V_0}{R_f}=0
3. Rearranging to solve for the output voltage V_0: -\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}\right) = \frac{V_0}{R_f} V_{0}=-R_{f}\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}\right)

Special Case: If all resistors are of equal value (R_f = R_1 = R_2 = R), the equation simplifies significantly: V_{0}=-R\left(\frac{V_1}{R}+\frac{V_2}{R}\right) V_{0}=-(V_{1}+V_{2}) In this case, the output is the inverted sum of the input voltages. The negative sign indicates a 180° phase inversion.

Subtractor (Difference Amplifier)

A subtractor, or difference amplifier, is a circuit that produces an output voltage proportional to the difference between two input voltages. The analysis of this circuit is most easily performed using the superposition theorem.

Output Voltage Derivation:

• Step 1: Consider Input Only We first calculate the output voltage contribution from V_1 by setting V_2 to zero (ground). The input V_1 passes through a voltage divider (R_2, R_3) to the non-inverting terminal. The circuit then acts as a non-inverting amplifier. The voltage at the non-inverting terminal, V_p, is: V_{p}=V_{1}\left(\frac{R_3}{R_2+R_3}\right) The output due to V_1, which we call V_{01}, is: V_{01}=V_{p}\left(1+\frac{R_f}{R_1}\right) = V_{1}\left(\frac{R_3}{R_2+R_3}\right)\left(1+\frac{R_f}{R_1}\right)
• Step 2: Consider Input Only Next, we calculate the output voltage contribution from V_2 by setting V_1 to zero. With the non-inverting terminal now at ground (through the resistor network), the circuit acts as a simple inverting amplifier for input V_2. The output due to V_2, which we call V_{02}, is: V_{02}=\left(-\frac{R_f}{R_1}\right)V_{2}
• Step 3: Combine the Results The total output voltage V_0 is the sum of the individual contributions: V_{0} = V_{01} + V_{02} V_{0}=V_{1}\left(\frac{R_3}{R_2+R_3}\right)\left(1+\frac{R_f}{R_1}\right)-\left(\frac{R_f}{R_1}\right)V_{2}

Special Case: If all resistors are of equal value (R_f = R_1 = R_2 = R_3 = R), the equation simplifies to a pure subtraction: V_{0}=V_{1}\left(\frac{R}{R+R}\right)\left(1+\frac{R}{R}\right)-\left(\frac{R}{R}\right)V_{2} V_{0}=V_{1}\left(\frac{1}{2}\right)(2)-V_{2} V_{0}=V_{1}-V_{2}

Building on these arithmetic principles, Op-Amps can also be configured to perform the calculus operations of differentiation and integration.

3.3 Differentiator and Integrator Circuits

Beyond basic arithmetic, the versatile Op-Amp can be configured to construct circuits that perform the fundamental mathematical operations of differentiation and integration. These circuits are critical components in analog signal processing, control systems, and waveform generation.

Differentiator

A differentiator is an electronic circuit that produces an output signal proportional to the first derivative (the rate of change) of its input signal. The circuit is a modification of the inverting amplifier where the input resistor is replaced by a capacitor.

Output Voltage Derivation:

1. The non-inverting terminal is grounded, so the inverting terminal is at a virtual ground (0V).
2. The nodal equation at the inverting terminal is: C\frac{\text{d}(0-V_{i})}{\text{d}t}+\frac{0-V_0}{R}=0
3. Rearranging the terms: -C\frac{\text{d}V_{i}}{\text{d}t}=\frac{V_0}{R}
4. Solving for the output voltage, V_0: V_{0}=-RC\frac{\text{d}V_{i}}{\text{d}t}

The output is the inverted, scaled derivative of the input. If the time constant RC is set to 1 second, the circuit provides unity gain for the derivative operation. The negative sign indicates a 180° phase difference between input and output.

Integrator

An integrator is an electronic circuit that produces an output signal that is the time integral of its input signal. This circuit is also based on the inverting amplifier configuration, but the feedback element is a capacitor, and the input element is a resistor.

Output Voltage Derivation:

1. The non-inverting terminal is at ground, so the inverting terminal is at a virtual ground (0V).
2. The nodal equation at the inverting terminal is: \frac{0-V_i}{R}+C\frac{\text{d}(0-V_{0})}{\text{d}t}=0
3. Rearranging the terms: \frac{-V_i}{R}=C\frac{\text{d}V_{0}}{\text{d}t} \frac{\text{d}V_{0}}{\text{d}t}=-\frac{V_i}{RC}
4. Integrating both sides with respect to time to find the output voltage, V_0: \int{d}V_{0}=\int\left(-\frac{V_i}{RC}\right){\text{d}t} V_{0}=-\frac{1}{RC}\int V_{i}{\text{d}t}

The output is the inverted, scaled integral of the input. If the time constant RC is set to 1 second, the circuit provides unity gain for the integration operation. The negative sign again indicates a 180° phase difference.

3.4 Converters of Electrical Quantities

In many electronic systems, such as sensor interfaces, instrumentation, and power management, it is necessary to convert between the fundamental electrical quantities of voltage and current. The Op-Amp provides a straightforward way to create circuits that perform these conversions accurately. The Voltage-to-Current (V-to-I) and Current-to-Voltage (I-to-V) converters are essential tools for this purpose.

Voltage to Current Converter (V-to-I Converter)

A Voltage-to-Current converter is an electronic circuit that produces an output current (I_0) that is directly proportional to its input voltage (V_i).

Circuit Derivation:

1. The input voltage V_i is applied to the non-inverting terminal.
2. Due to the virtual short concept, the voltage at the inverting terminal is also equal to V_i.
3. This voltage V_i appears across the resistor R_1. The other end of R_1 is connected to the load, which carries the output current I_0.
4. The nodal equation at the inverting terminal’s node is: \frac{V_i}{R_1}-I_{0}=0
5. Solving for the output current I_0: I_{0}=\frac{V_i}{R_1}

The output current is simply the input voltage divided by the resistance R_1. The gain of this circuit is the ratio of output current to input voltage, a quantity known as Transconductance. \text{Transconductance} = \frac{I_0}{V_i}=\frac{1}{R_1}

Current to Voltage Converter (I-to-V Converter)

A Current-to-Voltage converter is an electronic circuit that produces an output voltage (V_0) that is directly proportional to its input current (I_i). This circuit is often used to amplify the small currents generated by sensors like photodiodes.

Circuit Derivation:

1. The non-inverting terminal is connected to ground (0V).
2. Due to the virtual short concept, the inverting terminal is also at 0V.
3. The input current source I_i is connected to the inverting terminal. This current flows through the feedback resistor, R_f (since the Op-Amp’s infinite input impedance, described in Section 2.3, prevents any current from flowing into the inverting terminal).
4. The nodal equation at the inverting terminal’s node is: -I_{i}+\frac{0-V_0}{R_f}=0
5. Solving for the output voltage V_0: V_{0}=-R_{f}I_{i}

The output voltage is the product of the input current and the feedback resistance. The negative sign indicates a 180° phase difference. The gain of this circuit is the ratio of output voltage to input current, a quantity known as Transresistance. \text{Transresistance} = \frac{V_0}{I_i}=-R_{f}

These linear applications demonstrate the power of combining an Op-Amp with simple feedback networks. Next, we will explore applications where the Op-Amp is used in a non-linear fashion.